A TRACON Controller reported a departing Gulfstream did not respond to climb instruction and vectors and flew below the Minimum Vectoring Altitude.

Narrative

Aircraft X on the ZZZZZ departure climbing to 9;000 with VFR from the north going southbound restricted at or above 10;000 for departures; as well as an IFR aircraft from the west going eastbound (either level at 11;000 or climbing to 13;000; I don't remember). I was training and instructed developmental to turn Aircraft X right heading 360. Had Aircraft Y depart right after Aircraft X as well. Our attention was taken away with other aircraft; and developmental instructed Aircraft X direct their next fix and climb to 14;000. Aircraft X did not climb nor turned; so developmental turned them right heading 120 then direct next fix. Aircraft X read back the instruction but neither climbed nor turned and was level at 9;000 ft in the 9;200 ft MVA when I stepped in and had Aircraft X turn right heading 120 immediately. Today was the second day of the developmental training on the sector and the situation was complex for being ZZZ sector.The recommendation is more for myself: I'm very new at training; have never trained at ZZZ yet; and have only trained in radar 2 times before this event. I need to pay closer attention and step in a lot sooner when I notice the developmental falling behind.